H(?)lder Continuity And Differentiability Almost Everywhere Of (K₁,K₂)-Quasiregular Mappings

This paper deals with (K₁, K₂)-quasiregular mappings. Let f :Ω→Rⁿ be a(K₁, K₂)-quasiregular mapping, and∫_Ω|Df(x)|ⁿdx = Mⁿ < +∞.It is shown, by Morrey's Lemma and isoperimetric inequality, that every (K₁, K₂)-quasiregular mapping satisfies a H(o|¨)lder condition with exponentαon compact subsetsof its domain, whereAnd,if V is a subset of strictly contained inΩ,then for all x,y∈V,we have|f(x)-f(y)|≦L|x-y|^αwhere the constant L depends only on V , the constants K₁ and K₂, the dimension n, thedistance from V to the boundary ofΩand the constant M.It is also obtained that any(K₁, K₂)-quasiregular mapping is differentiable almost everywhere, and its differential isequal to Df(x).